Research on local sound field intensity control technique in metasurface based on deep neural networks

The use of tunable metasurface technology to realize the underwater tracking function of submarines, which is one of the hotspots and difficulties in submarine design. The structure-to-sound-field metasurface design approach is a highly iterative process based on trial and error. The process is cumbersome and inefficient. Therefore, an inverse design method was proposed based on parallel deep neural networks. The method took the global and local target sound field feature information as input and the metasurface physical structure parameters as output. The deep neural network was trained using a kernel loss function based on a radial basis kernel function, which established an inverse mapping relationship between the desired sound field to the metasurface physical structure parameters. Finally, the sound field intensity modulation at a localized target range was achieved. The results indicated that within the regulated target range, this method achieved an average prediction error of less than 5 dB for 92.9% of the sample data.


Introduction
Acoustic super-surface materials, utilizing the technology of modifying material properties based on the generalized Snell's law, possess exceptional acoustic control abilities at subwavelength scales.Therefore, they have emerged as research hotspots in areas such as acoustic lenses, noise reduction, and acoustic stealth [1][2][3][4].Pentamode acoustic metasurface can further compensate for the narrow-band limitations of traditional metasurfaces [5,6].So far, researchers have achieved some sound field modulation works through the use of various structural designs such as Helmholtz resonators, coiled channels, mazes, and cavities [7][8][9][10][11][12][13]. Also, the researchers have utilized combinations of different materials, such as a combination of water and silicone rubber or polyurethane composites, to achieve the goal of reflective sound field modulation [14,15], other researchers use bottom-up inversion optimization algorithms to design metasurfaces [16][17][18][19].While these phase mutation methods can achieve acoustic field modulation [20][21][22][23][24][25], it is noteworthy that all of the aforementioned methods have been studied based on the forward model of the sound field and finite element simulation.This research process requires a substantial amount of time from the researchers.
Considering the limitations of conventional metasurface research methods, machine learning methods have been applied to the acoustic metasurface inverse design process, which enabled the optimization of structural parameters.Some results have been achieved [26][27][28].Among them, Zhao et al. used a convolutional neural network model to establish a mapping of the local acoustic field to the phase gradient of the metasurface to achieve regional control of the local acoustic field [27].Li et al. proposed a tandem neural network approach to reverseengineer the phase of a metasurface unit such that the energy loss of an acoustic wave in the return direction is greater than 10 dB [28].Long, Chen et al. used genetic algorithms to respectively design metasurface structures for sound absorption [26,29].Li, Lin, et al. have respectively used machine learning for encoding metasurfaces to enable the modulation of the sound field by arranging these logical units into specific sequences [30,31].These studies have taken advantage of the benefits of machine learning methods for model construction, which can help weaken complex physical mechanisms and reduce the need for model accuracy.This shows that the addition of deep learning is relevant to metasurface modulation techniques.
This paper introduces a novel metasurface inverse design method leveraging parallel deep neural networks (PDNN).The method respectively extracts the key information of the acoustic field and the metasurface as the input and output of the PDNN network.With the help of the kernel loss function and the constraint performance provided by the constraint network, it establishes an inverse mapping relationship between the target acoustic field and the physical structure parameters of the super-surface.Model validation show that this method can realize the regulation of local sound field intensity.This may be a novel way to achieve stealth for submarine vehicles.

Physical model of the metasurface local sound field
Turing the process of realizing intensity modulation of the target acoustic field, a physical model of the metasurface local sound field was used to acquire the dataset.The pentamode metasurface was chosen for sound field simulation because of its advantages of impedance matching with the ambient medium and wide frequency [32].For a pentamode metasurface based on the generalized Snell's law, the material density distribution ρ(x) is the decisive parameter affecting the reflected acoustic field.When the sound wave is vertically incident on the pentamode metasurface with the acoustic velocity c 0 , its ideal density distribution ρ(x) satisfies Eq (1) [33]: where L is the length of the metasurface, C 0 is the integration constant, θ r is the reflection angle, ρ 0 is the density of incident medium, h is the normal thickness of metasurface, and x is the position.Artificial periodic structures cannot realize a continuous material density distribution on the theoretical metasurface.To approximate this continuous distribution, the theoretical metasurface can be discretized into n cells along the length (i = 1, 2, . .., n).The density of each discrete cell is characterized by the density ρ i at its center position [34,35].
In this paper, we started from the idea of parametric modeling without structural constraints on the metasurface structural units.Simplified parameters were used instead of metasurface structural units.With the method of unit combination, the phase mutation was adjusted at the same time to realize the local tuning of the acoustic field.The hypersurface has a normal thickness of 0.12m and a length of 1m.In the example of a sonar-detecting submarine shown in Fig 1(A), the incident acoustic wave can be viewed as a plane wave.When the incident acoustic wave contacts the surface of the submersible, the main reflected acoustic field is adjusted from the 90˚direction to the other direction through the acoustic field that information: https://github.com/bjhkbj/dataset. git.

Competing interests:
The authors declare no conflict of interest.modulation technique.The intensity of the acoustic field in the return direction is also changed.Therefore, we established the physical model of the sound field following the approach illustrated in Fig 1 (B).A plane wave incident vertically underwater is used as the background field.The metasurface covered the backing plate surface and consisted of n metasurface structural units arranged along the x-positive direction.When a plane wave is incident on the metasurface in the y-reverse direction, the reflected waves generated by the n metasurface structural units interacting with each other make up the entire reflected acoustic field.The physical structural parameters of each structural unit could be different.In this paper, the physical structural parameters of each unit were obtained according to a gradient arrangement, which satisfied the requirements of the intensity characteristics of the desired sound field distribution, set n to 25, so the length of each hypersurface structural unit is 0.04m.

Extraction of sound field features
In order to predicted the intensity of the target sound field, this paper drew on the idea of multi-scale that was to extract the global and local feature information of the sound field.The goal is the modulation of the local acoustic field, but the entire reflected acoustic field is a joint action of all metasurface structural units.In particular, the coupling relationship between individual structural units will have a significant impact on the reflected acoustic field characteristics.This coupling relationship will greatly increase the complexity of the model [36].Therefore, when using local sound field intensity values as parallel deep neural network inputs, it was necessary to include feature information of the global sound field to constrain this prediction process.
The prediction of target sound field strength required the selection of global and local feature information.According to Fig 2, it can be seen that the main change features of the reflected sound field are concentrated in the vicinity of the main reflection angle, the wave crest and trough, so the selection of global feature information can be extracted at the main change features.The local sound field intensity information was the value of the sound field intensity within the selected tuning target.After the global and local sound field feature information was extracted, it was used as an input to the parallel deep neural network.

Network model building
In this paper, a parallel deep neural network based on a fully connected architecture was used to predict the physical structural parameters of metasurface structural units.The model inputs were the extracted global and local feature information.The outputs were the density ρ and the gradient value g of the first structural unit.The density distribution of the entire  In the loss function selection, the MSE function is usually adopted as the loss function.However, the MSE loss function cannot accurately assess the nonlinear characteristics of the error and is sensitive to outliers.This problem can be solved by modifying the loss function using the radial basis kernel function.The modified loss function L Kernel-MSE-Single can be written as Eq (2) [37]: where N was the number of samples and σ was the parameter of the loss function itself, y t and ŷ t respectively represented true and predicted values.We set φ = (yŷ) 2 and λ = 2σ 2 , where y was the true value and ŷ was the predicted value, so φ was the squared error between the true value and the predicted value.According to Eq (2) and Fig 3, the final loss function L K-MSE expression was shown in Eq (3): where φ t1 and φ t2 respectively represented the squared errors of the true and predicted values of the constraint network, the prediction network.λ 1 and λ 2 were respectively the number of input features for the constraint and prediction networks.The modified loss function computed the gradient to the network parameters and completed the update to the network parameters.

Dataset preparation and setup parameters
The specific composition process of the dataset was shown in Fig 4, which consisted of two parts: labels and inputs.The labeling part consisted of the first metasurface unit density ρ and gradient value g.We randomly generated 30,000 sets of first block metasurface structural unit densities ρ and gradient values g.The metasurface density distributions could be calculated from the generated data, which were inputted into a physical model of the acoustic field to derive the corresponding acoustic field intensity distributions.The input parts were global sound field feature information and local sound field intensity information, which could be obtained by feature extraction of the sound field intensity distribution.In Fig 1(A), the reflected wave will be reflected along the echo direction (y reverse direction) when there is no metasurface, so the global sound field range could be set to 0˚~180˚.The energy of the reflected wave is mainly concentrated in the direction of the echo [38].Therefore, we set the target regulation interval as 85˚~95˚.When extracting the sound field intensity values within the tuning target range, we sampled at 0.5˚equal intervals with a dimension of 1×21.The sound field in the range of 0˚~180˚was divided into 6 intervals at equal step.The rules for extracting global sound field features were as follows: Step 1, Within each interval, select: 1.The sound field intensity values and angles of the maximum crest and two adjacent points on each side are required, 2. The sound field intensity values and angles of the minimum trough trough and two adjacent points on each side are required.
Step 2: Within the global acoustic field, the act of selection: 1.The intensity value of the main reflection angle sound field, 2. The number of maximum peaks.
The above features were selected to characterize the global feature information of the sound field with a dimension of 1×122.
The number of neurons for the neural network model was set as shown in Table 1.The relevant parameters for training the network model were set as shown in Table 2.

Network model training
In network model training, the constraint performance of the constraint network directly affects the final network prediction results.If the constraint network weight α was too large means that the network model prediction results were more biased towards the global sound field distribution.Thus, the target sound field intensity prediction was not accurate enough.If the constraint network weight α was small means that the link between the target sound field intensity and the global sound field was weakened, and the coupling relationship between multiple structural units couldn't be learned during the model training process.The prediction results will also be inaccurate.Therefore, a key point in realizing local sound field intensity prediction was to determine the optimal weighting of the constraint network in the overall neural network by adjusting the weighting factors.
In this paper, the value of weight factor α was discussed.Under the unchanged conditions of the aforementioned parameter settings, model training and generalization ability verification were performed with different values of α (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9).The    In the above equation e is the mean error, P is the true value and P(ˆ) represents the predicted value.

Loss function comparison model validation
This study introduced a comparative model validation between the K-MSE loss function and other commonly used loss functions which were SmoothL1, Quantile, Huber, MAE, and MSE.Their specific formulas were given as Eqs ( 5)- (9).Comparative model validation could demonstrate that the K-MSE loss function can help to construct a mapping relationship between  1 and 2, while α = 0.4.Only the loss function changed throughout the training process.where y t was the true value, ŷ t was the model predicted value, and N was the number of samples.In Eq (6), q was the quantile, q = 0.8.In Eq (7), δ was the hyperparameter of L Huber , δ = 0.5.The generalization ability of the network model under the six loss functions was also verified using a randomly sampled set of 1000 data sets that were not involved in training.The performance of the network model with different loss functions is shown in Table 3.The K-MSE loss function achieves P percentage of 92.9% when e is less than 5 dB in the target control range.It also has the smallest loss function value and the fastest convergence rate, which indicates its optimal optimization performance for the prediction network.The predicted sound field distribution for the same set of randomly selected parameters is shown in       sound field intensity curve obtained by this method closely follows the variation trend of the simulated sound field intensity curve.Additionally, it achieves a percentage of 92.9% for the sample data in which the average error between the predicted sound field intensity values and the true values falls within the specified control target range of less than 5 dB.
Fig 12 (A)-12(E) demonstrates that the errors in the main reflection angles of the predicted and real sound fields are within 15˚.The envelope trend of the sound field intensity distribution of the predicted sound field and the real sound field are similar, but the number of peaks and valleys

Fig 11 .
Fig 11.The average error between the true value of the local sound field intensity and the predicted value.https://doi.org/10.1371/journal.pone.0301211.g011 This paper proposed a metasurface inverse design method based on parallel deep neural networks (PDNN).The method respectively established a prediction network with local target acoustic field intensity and metasurface physical structure parameters as input and output.The global acoustic field features were used as inputs to the constraint network.The weight of the constraint network in the whole PDNN network was adjusted by adjusting the weight factor.The loss function based on the radial basis kernel function was used to train the whole network model and construct the mapping relationship from the desired sound field to the metasurface physical structure parameters.The predicted metasurface structural unit parameters could be derived from the desired acoustic field.Ultimately, the modulation of local acoustic field intensity was realized.The model validation results demonstrate that the predicted